The aim of this paper is to study the existence of various types of peak solutions for an elliptic system of FitzHugh-Nagumo type. We prove that the system has a single peak solution, which concentrates near the boundary of the domain. Under some extra assumptions, we also construct multi-peak solutions with all the peaks near the boundary, and a single peak solution with its peak near an interior point of the domain.

We show that the critical problem {-∆υ + λυ = υ²*⁻¹ + αυq⁻¹, υ > 0 in Ω, ∂υ∕∂ν = 0 on ∂Ω, 2 < q < 2* = 2N/(N − 2), has no positive solutions concentrating, as λ → ∞, at interior points of Ω if a = 0, but for a class of symmetric domains Ω, the problem with α > 0 has solutions concentrating at interior points of Ω.

We prove that certain weakly nonlinear elliptic equations have many solutions when a paragraph is large. The nonlinearity grows superlinearly for y positive but grows linearly for y negative.

We consider a minimization problem associated with the elliptic systems of FitzHugh–Nagumo type and prove that the minimizer of this minimization problem has not only a boundary layer, but also may oscillate in a set of positive measure.

In this paper, we study existence of solutions with boundary layer and peaks for an elliptic problem. We prove that the problem has a mountain-pass-type solution, which has a boundary layer and a single peak near the boundary. Moreover, we also study the existence of solutions with interior peak.

In this paper, we shall construct multibump solutions for an elliptic problem on this expanding domain, such that all the local maximum points of the solution are close to the set...

This parabolic system in one space dimension is a simplification of the original Hodgkin-Huxley nerve conduction equations. Here u denotes an activator and v acts as its inhibitor. This system has also been studied in many applied areas. The readers can find more references in for the background of the systems of FitzHugh--Nagumo type. Here we mention some early results on the systems of FitzHugh--Nagumo type, obtained by Klaasen and Troy, Klaasen and Mitidieri and DeFigueiredo and Mitidieri.

In this paper, we consider the following elliptic problem... where a(x) is a continuous function satisfying 0 < a(x)<1 for x∈...,Ω is a bounded domain in R^N with smooth boundary, ε > 0 is a small number. A solution of (1.1) can be interpreted as a steady state solution of the corresponding problem: ut=ε²Δu+f(x,u), where f(x,t)=t(t-a(x))(1-t), which arises in a number of places such as population genetics [24,33]. The readers can refer to [22] for more background material for problem (1.1).

We consider nonlinear elliptic equations with small diffusion and Dirichlet boundary conditions. We construct changing sign solutions with peaks close to the boundary and consider the location of the peak.

In this paper, we construct three types of symmetric peaked solutions for a Neumann problem involving critical Sobolev exponent: the interior peaked solution, the boundary peaked solution and the interior-boundary peaked solution.